Noether Theorems and Discrete Variational Integrators in Field Theory

• Authors: Li-Li Xia , Li-Qun Chen , Chang-Xin Liu
• Languages of publication: EN

Abstracts

EN

The discrete analogue of the Noether-type identities in field theory is investigated by means of the difference discrete variational principle in which the difference is regarded as an entire geometric object. The discrete counterparts of the Noether theorems are obtained. It is proved that there exists the discrete version of the Noether conservation law in field theory. The discretization for the nonlinear Schrödinger equation is presented to illustrate the results.

Keywords

EN

02.20.Sv; 11.30.-j

Discipline

• 11.30.-j: Symmetry and conservation laws(see also 02.20.-a Group theory)
• 02.20.Sv: Lie algebras of Lie groups

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2015
• Volume: 127
• Issue: 3
• Pages: 669-673

Dates

published

2015-03

received

2014-05-01

(unknown)

2015-01-31

Contributors

author

Li-Li Xia

• Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
• Department of Physics, Henan Institute of Education, Zhengzhou 450046, China

author

Li-Qun Chen

• Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
• Department of Mechanics, Shanghai University, Shanghai 200444, China
• Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai 200072, China

author

Chang-Xin Liu

• Department of Physics, Henan Institute of Education, Zhengzhou 450046, China

References

• [1] G.D. Birkhoff,Proc. Am. Acad. Arts Sci. 49, 521 (1913), doi: 10.2307/20025482
• [2] J.A. Cadzow, Int. J. Control 11, 393 (1970), doi: 10.1080/00207177008905922
• [3] A.E. Noether, Nachr. König. Gesell. Wissen. Göttingen, Math. Phys. KI. 2, 235 (1918), doi: 10.1007/978-3-642-39990-9_13
• [4] S. Maeda, Math. Japan 23, 587 (1979)
• [5] V. Dorodnitsyn, R. Kozlov, J. Phys. A Math. Theor. 42, 454007 (2009), doi: 10.1088/1751-8113/42/45/454007
• [6] J.L. Fu, G.D. Dai, J. Salvador, Y.F. Tang, Chin. Phys. 16, 570 (2007)
• [7] J.L. Fu, B.Y. Chen, F.P. Xie, Chin. Phys. 17, 4354 (2008)
• [8] J.L. Fu, X.W. Li, C.R. Li, W.J. Zhao, B.Y. Chen, Sci. China Phys. Mech. 53, 1699 (2010), doi: 10.1007/s11433-010-4075-1
• [9] P.P. Cai, J.L. Fu, Y.X. Guo, Sci. China Phys. Mech. 56, 1017 (2013), doi: 10.1007/s11433-013-5065-x
• [10] D.D.S. Djukic, B.D. Vujanovic, Acta Mech. 23, 17 (1975), doi: 10.1007/BF01177666
• [11] W. Sarlet, F. Cantrijn, SIAM Rev. 23, 467 (1981), doi: 10.1137/1023098
• [12] F.X. Mei, Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems, Science Press, Beijing 1999
• [13] Z.P. Li, Int. J. Theor. Phys. 26, 853 (1987)
• [14] Z.P. Li, Int. J. Theor. Phys. 32, 201 (1993), doi: 10.1007/BF00674405
• [15] R. Arens, Commun. Math. Phys. 90, 527 (1983), doi: 10.1007/BF01216184
• [16] J.E. Marsden, G.W. Patrick, S. Shkoller, Commun. Math. Phys. 199, 351 (1998), doi: 10.1007/s002200050505
• [17] H.Y. Guo, Y.Q. Li, K. Wu, S.K. Wang, Commun. Theor. Phys. 37, 1 (2002)
• [18] J.B. Chen, Lett. Math. Phys. 61, 63 (2002)
• [19] A. Trautman, Commun. Math. Phys. 6, 248 (1967), doi: 10.1007/BF01646018
• [20] V. Dorodnitsyn, R. Kozlov, P. Winternitz, J. Math. Phys. 45, 336 (2004), doi: 10.1063/1.1625418
• [21] J.L. Fu, B.Y. Chen, Mod. Phys. Lett. B 23, 1315 (2009), doi: 10.1142/S0217984909019417
• [22] R. Scharf, A. Bishop, Phys. Rev. A 43, 6535 (1991), doi: 10.1103/PhysRevA.43.6535

Identifiers

DOI

10.12693/APhysPolA.127.669